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**Model theory.
3rd rev. ed.**
*(English)*
Zbl 0697.03022

Studies in Logic and the Foundations of Mathematics, 73. Amsterdam etc.: North-Holland. xvi, 650 p. $ 118.00; Dfl. 230.00 (1990).

This book grew out of a number of graduate courses in model theory that the authors taught at UCLA and Wisconsin. It was tested in classes and completely rewritten in 1967-68. Major changes were again made in 1971- 72. In 1973 the first edition appeared in print (see Zbl 0276.02032). It covered most of first-order model theory and of its applications to algebra and model theory. Soon it became the standard textbook for model theory. It is now widely used as a beginning graduate textbook and also as a reference work. The second edition appeared already in 1976 with only minor changes.

Here in the third edition, some new topics have been added, which now belong to a first graduate course. So four new sections have been added, also new exercises, which are of great importance for understanding the text. The appendix is updated and the book contains a list of additioal references.

Now we come to the new sections. Section 2.4 introduces recursively saturated models. They can be used to simplify results in model theory which had been obtained earlier with the help of saturated models. As an application an easy direct proof of the Robinson Consistency Theorem is given. Recursively saturated models are also used to simplify the proof of the Vaught two-cardinal theorem. In section 2.5 the Lindström characterization of first-order logic is presented. It states that first- order logic is the only logic for which the Compactness Theorem and the Downward Löwenheim Skolem Theorem hold. The new section 3.5 deals with model completeness. In the earlier editions model completeness was dealt with in section 3.1. Here the joint embedding property and the amalgamation property are investigated. The last new section 4.4 deals with nonstandard universes. Here two approaches are developed: superstructures and internal set theory. This gives the basis to apply results from model theory to nonstandard analysis.

Here in the third edition, some new topics have been added, which now belong to a first graduate course. So four new sections have been added, also new exercises, which are of great importance for understanding the text. The appendix is updated and the book contains a list of additioal references.

Now we come to the new sections. Section 2.4 introduces recursively saturated models. They can be used to simplify results in model theory which had been obtained earlier with the help of saturated models. As an application an easy direct proof of the Robinson Consistency Theorem is given. Recursively saturated models are also used to simplify the proof of the Vaught two-cardinal theorem. In section 2.5 the Lindström characterization of first-order logic is presented. It states that first- order logic is the only logic for which the Compactness Theorem and the Downward Löwenheim Skolem Theorem hold. The new section 3.5 deals with model completeness. In the earlier editions model completeness was dealt with in section 3.1. Here the joint embedding property and the amalgamation property are investigated. The last new section 4.4 deals with nonstandard universes. Here two approaches are developed: superstructures and internal set theory. This gives the basis to apply results from model theory to nonstandard analysis.

Reviewer: M.Weese

### MSC:

03Cxx | Model theory |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03H99 | Nonstandard models |